1. **Stating the problem:** We want to factor a quadratic polynomial of the form $ax^2 + bx + c$ using the quadratic formula. 2. **Formula used:** The quadratic formula to find roots of $ax^2 + bx + c = 0$ is: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Important rules:** - The discriminant $\Delta = b^2 - 4ac$ determines the nature of roots. - If $\Delta > 0$, two distinct real roots. - If $\Delta = 0$, one real root (repeated). - If $\Delta < 0$, no real roots (complex roots). 4. **Steps to factor:** - Find roots $r_1$ and $r_2$ using the quadratic formula. - Then factor the polynomial as: $$ax^2 + bx + c = a(x - r_1)(x - r_2)$$ 5. **Example:** Factor $2x^2 + 5x - 3$. - Calculate discriminant: $$\Delta = 5^2 - 4 \times 2 \times (-3) = 25 + 24 = 49$$ - Find roots: $$x = \frac{-5 \pm \sqrt{49}}{2 \times 2} = \frac{-5 \pm 7}{4}$$ - Roots: $$r_1 = \frac{-5 + 7}{4} = \frac{2}{4} = \frac{1}{2}$$ $$r_2 = \frac{-5 - 7}{4} = \frac{-12}{4} = -3$$ - Factor: $$2x^2 + 5x - 3 = 2(x - \frac{1}{2})(x + 3)$$ - To avoid fractions, multiply inside factors: $$= 2 \left(x - \frac{1}{2}\right)(x + 3) = (2x - 1)(x + 3)$$ 6. **Summary:** Use the quadratic formula to find roots, then write the polynomial as $a(x - r_1)(x - r_2)$ and simplify if needed.